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ΣL(2,𝔽₈) #13094882

order	1512
isoclinism	isoclinic class of ΣL(2,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₁

Aliases

projective general semilinear	PΓL(2,𝔽₈)
special semilinear	ΣL(2,𝔽₈)
projective special semilinear	PΣL(2,𝔽₈)
unambiguous semidirect product	SL(2,𝔽₈)⋊C₃

Splittings

SL(2,𝔽₈) ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, socle, cosocle, hypocenter	SL(2,𝔽₈)	C₃
LCS	G ▷ SL(2,𝔽₈)
UCS	C₁

25 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 9
transitive degree	μ_t(G) = 9

A minimal faithful permutation representation — 2 generators on 9 points:

(1,9,7,6,5,4,2);(1,9,7,5,4,3)(6,8)

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁
size	1	56	63	84	84	168	168	168	216	252	252
order	1	3	2	3	3	9	9	9	7	6	6
χ₁	1	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₃	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$
χ₄	7	-2	-1	1	1	1	1	1	0	-1	-1
χ₅	7	-2	-1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	0	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$
χ₆	7	-2	-1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	0	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$
χ₇	8	-1	0	2	2	-1	-1	-1	1	0	0
χ₈	8	-1	0	$- 1 + i \sqrt{3}$	$- 1 - i \sqrt{3}$	-1	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	1	0	0
χ₉	8	-1	0	$- 1 - i \sqrt{3}$	$- 1 + i \sqrt{3}$	-1	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	1	0	0
χ₁₀	21	3	-3	0	0	0	0	0	0	0	0
χ₁₁	27	0	3	0	0	0	0	0	-1	0	0

Here i is the imaginary unit (i² = −1).

Relationships

maximal subgroup of	ΓL(2,𝔽₈)
maximal quotient of	ΓL(2,𝔽₈)
automorphism group of	SL(2,𝔽₈)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1512, 779);